Permutations and Combinations - Gr8AmbitionZ

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    Permutations and

    Combinations Important points to Remember

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    SUMMARY OF PERMUTATIONS & COMBINATIONSFormulas:No. of ways of selecting r objects out of n objects = nCrNo. of ways of arranging n objects = n!

    No. of ways of arranging n objects with a identical & b identical = n!

    a! b!No. of ways of arranging r objects out of n objects = n(n1)(n2) ... (n (r 1)) = nPr =

    nCr r!No. of ways of arranging n objects in a circle = (n 1)!

    Choosing PeopleA team of 4 is to be chosen from a group consisting of Anne and Bob and 4 other people. In how many ways can this be done if(i) there are no restrictions?(ii) Anne must be in the team?(iii) Anne and Bob must both be in the team?(iv) at most one of Anne and Bob in the team?(v) Anne or Bob or both are in the team?

    (i) No. of ways of choosing 4 people = 6C4 = 15(ii) No. of ways of choosing the other 3 people = 5C3 = 10(iii) No. of ways of choosing the other 2 people = 4C2 = 6(iv) Total no. of ways no. of ways with Anne & Bob both in the team = 15 6 = 9(v) No. of ways with Anne in the team + no. of ways with Bob in the team no. of ways with both in the team = 10 + 10 6 = 14

    Choosing from Different Types of People (e.g. Boys & Girls)A team of 3 is to be chosen from a group of 3 boys and 4 girls. How many ways can this be done if(i) there are no restrictions?(ii) there must be exactly 1 boy?(iii) there must be at least 1 boy?(iv) there must be at least 1 boy and at least 1 girl?

    (i) No. of ways of choosing 4 people = 7C3 = 35(ii) No. of ways of choosing 1 boy and 2 girls = 3C1

    4C2 = 18(iii) No. of ways = Total no. of ways no. of ways with no boys = 35 4C3 = 35 4 = 31

    Note: It is wrong to say no. of ways = 3C16C2 = 45

    (iv) No. of ways = Total no. of ways no. of ways with no boys no. of ways with no girls= 35 4C3

    3C3 = 35 4 1 = 30Note: It is wrong to say no. of ways = 3C1

    4C15C1 = 60

    Choosing People to form GroupsFind the no. of ways in which 6 people can be divided into (i) two groups consisting of 4 & 2 people, (ii) two groups consisting of 3 people each.(iii) group 1 and group 2, with 3 people in each group.

    (i) No. of ways = 6C42C2 = 15

    (ii) No. of ways = 6C3

    3C32! = 10 Note: Divide by 2! since the 2 groups are of equal size.

    (iii) No. of ways = 6C33C3 = 20 Note: Dont divide by 2! since the 2 groups are labeled.

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    Choosing Letters, Balls or other Identical ObjectsSeven cards each bear a single letter, which together spells the word MINIMUM. Three cards are to be selected. The order of selection is disregarded. Find the number of different selections.

    Case 1: No. of ways of choosing 3 identical letters = no. of ways of choosing MMM = 1Case 2: No. of ways of choosing MM + 1 other letter = no. of ways of choosing I, N, U = 3C1 = 3Case 3: No. of ways of choosing I I + 1 other letter = no. of ways of choosing M, N, U = 3C1 = 3Case 4: No. of ways of choosing 3 different letters = no. of ways of choosing M, I, N, U = 4C3 = 4total no. of selections = 1 + 3 + 3 + 4 = 11

    Arranging PeopleAnne and Bob and 2 other people are to sit in a row. How many ways can this be done if(i) there are no restrictions?(ii) Anne must sit on the left and Bob on the right?(iii) Anne and Bob must sit together?(iv) Anne and Bob must be separate?

    (i) No. of ways of arranging 4 people = 4! = 24(ii) No. of ways of arranging the other 2 people = 2! = 2(iii) Treat Anne and Bob as 1 item.

    No. of ways of arranging 3 items no. of ways of arranging Anne & Bob = 3! 2! = 12(iv) No. of ways = 24 12 = 12

    Arranging Different Types of People (e.g. Boys & Girls)In how many ways can 3 boys & 3 girls be arranged in a row if(i) there are no restrictions? (ii) the 1st person on the left is a boy? (iii) the person on each end is a boy? (iv) the boys are together?(v) the boys are separate?

    (i) No. of ways of arranging 6 people = 6! = 720(ii) No. of arrangements = 3C1 no. of ways of arranging the other 5 people = 3C1 5! = 360(iii) No. of arrangements = 3C2 2! no. of ways of arranging the other 4 people

    = 3C2 2! 4! = 144(iv) Treat the 3 boys as 1 item.

    No. of ways of arranging 4 items = 4!The boys can be arranged among themselves in 3! waysno. of arrangements = 4! 3! = 144

    (v) G1 G2 G3

    The 3 girls can be arranged in 3! ways.From the 4 spaces, choose 3 places for the boys in 4C3 ways, and arrange them in 3! ways.no. of arrangements = 3! 4C3 3! = 144Note: It is wrong to subtract no. of ways where boys are together from the total no. of ways.

    Arranging Letters, Balls or other Identical Objects(a) Find the number of arrangements of all 7 letters of the word MINIMUM in which

    (i) there are no restrictions.(ii) the 3 letters M are next to each other(iii) the 3 letters M are separate

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    (iv) the first letter is M(v) the first & last letters are M(vi) the first letter is M or the last letter is M or both

    (b) Find the number of 4letter codewords that can be made from the letters of the word MINIMUM.

    (ai) No. of arrangements = 7!

    3! 2! = 420

    (ii) Treat MMM as 1 item.

    No. of arrangements of MMM, I, N, I, U =5!2! = 60

    (iii) I N I U

    The letters I, N, I, U can be arranged in 4!2! ways

    From the 5 spaces, choose 3 places for the 3 Ms in 5C3 ways.

    no. of arrangements = 4!2!

    5C3 = 120

    (iv) M _ _ _ _ _ _

    No. of ways of arranging I, N, I, M, U, M = 6!

    2! 2! = 180

    (v) M _ _ _ _ _ M

    No. of ways of arranging I, N, I, M, U = 5!2! = 60

    (vi) No. of arrangements = 180 + 180 60 = 300

    (b) Case 1: M, M, M & 1 other letter: No. of words = 3C14!3! = 12

    Case 2: M, M, I, I: No. of words = 4!

    2! 2! = 6

    Case 3: M, M & 2 different letters: No. of words = 3C24!2! = 36

    Case 4: I, I & 2 different letters: No. of words = 3C24!2! = 36

    Case 5: 4 different letters: No. of words = 4! = 24Total no. of codewords = 12 + 6 + 36 + 36 + 24 = 114

    Arranging People Around a Table4 men and 3 women are to sit at a round table. Find the number of ways of arranging them if(i) there are no restrictions.(ii) the 3 women must sit together.(iii) the 3 women must be separate.(iv) the 3 women must be separate and the seats are numbered 1 to 7.(i) No. of ways of arranging 7 people around a table = (7 1)! = 720(ii) Treat the 3 women as 1 item.

    No. of ways of arranging 5 items around a table = (5 1)!The 3 women can be arranged among themselves in 3! waysno. of arrangements = (5 1)! 3! = 144

    (iii) Let the 4 men sit down first in (4 1)! ways.From the 4 spaces, choose 3 places for the women in 4C3 ways.Arrange the 3 women in 3! ways.total no. of arrangements = (4 1)! 4C3 3! = 144

    (iv) No. of arrangements = 144 7 = 1008

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